Heat Conduction Analysis of a Functionally Graded Cylinder in Transient Regime with Chebyshev Pseudospectral Method

Abstract

Transient analysis of functionally graded infinite length hollow cylinder is performed using non-Fourier hyperbolic heat conduction model. Except the thermal relaxation time, the material properties are assumed to change exponentially in the radial direction. Since an infinite-length cylinder is considered, the effects at the upper and lower ends of the cylinder are neglected. Under these conditions, a partial differential equation with variable coefficients is obtained which is only depend on the radial variable in the space direction. By applying Laplace transform to this differential equation, the linear ordinary differential equation obtained in Laplace space is solved numerically by using Chebyshev Pseudospectral method and the solution in time space is obtained by using Modified Durbin Inverse Transformation Method. The transient dynamic responses of temperature and heat flux are investigated by taking a special metal-ceramic mixture and examining various relative temperature changes. The behavior of temperature distribution and heat flux at different times is shown in the form of graphs. The solutions available in the literature are used to demonstrate the accuracy of this combined method. The combined method used in this study has been shown to be a well structured, simple, effective.

Keywords

• Hyperbolic Heat Conduction in Cylinders
• Functionally Graded Material
• Laplace Transform
• Chebyshev Pseudospectral Method
• Durbin Inverse Transform Method

Fonksiyonel Derecelendirilmiş Silindirin Isı İletiminin Chebyshev Pseudospektral Yöntemi İle Geçici Rejimdeki Analizi

Abstract

Fourier olmayan hiperbolik ısı iletim modeli kullanılarak fonksiyonel derecelendirilmiş sonsuz uzunlukta içi boş silindirin geçici rejimdeki analizi yapılmıştır. Termal gevşeme süresi dışında malzeme özelliklerinin radyal yönde üstel olarak değiştiği kabul edilmiştir. Sonsuz uzunlukta bir silindir ele alındığından silindirin alt ve üst ucundaki etkiler ihmal edilmiştir. Bu koşullar altında, uzay yönünde sadece radyal değişkene bağlı değişken katsayılı kısmi diferansiyel denklem elde edilir. Bu diferansiyel denkleme Laplace dönüşümü uygulanarak, Laplace uzayında zamandan bağımsız elde edilen lineer adi diferansiyel denklem, Chebyshev Pseudospektral yöntemi kullanılarak sayısal olarak çözülüp, Modifiye Edilmiş Durbin Ters Dönüşüm Yöntemi kullanılarak zaman uzayındaki çözüm elde edilir. Sıcaklık ve ısı akısının geçici dinamik tepkileri, özel bir metal-seramik karışımı alınarak ve çeşitli göreceli sıcaklık değişikliklerine göre incelenmiştir. Farklı zamanlarda sıcaklık dağılımı ve ısı akısının davranışı farklı şekiller üzerinde gösterilmiştir. Bu birleştirilmiş yöntemin doğruluğunu göstermek için literatürde mevcut olan çözümler kullanılmıştır. Bu çalışmada kullanılan birleştirilmiş yöntemin, iyi yapılandırılmış, basit, etkili bir yöntem olduğu gösterilmiştir.

Keywords

• Silindirlerde Hiperbolik Isı İletimi
• Fonksiyonel Derecelendirilmiş Malzeme
• Laplace Dönüşümü
• Chebyshev Pseudospektral Yöntemi
• Durbin Ters Dönüşüm Yöntemi

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